How to train for recent Silver questions?

General Questions
#1

It seems to me that recent Silver problems require you to make very high-level deductions about the problem at hand. For example, in last competition’s problem, “Farmer John’s Favorite Operation,” the solution cites convex functions as a vital step in getting full credit, and I hadn’t heard of or seen convex functions used in any of the practice problems I did to prepare for the contest.

I believe I am proficient in all of the concepts presented in the USACO guide for Silver, but I’m not sure how to learn to make the observations required to solve new Silver problems. Even in similar-difficulty problems I practice, their individual solutions seem so specific that I don’t understand how I can apply my knowledge of them to other questions.

Overall, I’m unsure how I should train to solve recent Silver problems; I don’t feel as though I am able to make the deductions required for more than one or two subtasks at all, much less within the 4 hour time limit. My intention is not to complain about the difficulty of Silver, but to better understand how I can learn the concepts needed to overcome it. While I am able to solve older Silver questions without much trouble, I’m unable to wrap my head around the modern ones.

Thank you for your time.

#2

Yeah the editorial can be rather intimidating if you haven’t heard of convex functions before.

That being said, I don’t quite agree with this?

  • Sure, if you know g is convex in approach 1, then it helps since it automatically implies g is decreasing, then equal on a range, then increasing. But it’s possible to observe this without having heard of convex functions beforehand.
  • In approach 2, we use a fact about concave function to show its correctness, but you can alternatively just assume that the observation is true without proving it.
  • A third possible approach (which doesn’t use convexity) is to express g as a piecewise linear function, then it can be solved similarly as this USACO Silver problem: https://usaco.org/current/data/sol_teleport_silver_feb18.html (which admittedly was the most challenging problem of that Silver contest …).
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#3

Some additional practice problems:

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